Skip to main content
SpringerLink
Log in

On the bounds and achievability about the ODPC of \(\mathcal{GRM }(2,m)^*\) over prime fields for increasing message length

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

Abstract

The optimum distance profiles of linear block codes were studied for increasing or decreasing message length while keeping the minimum distances as large as possible, especially for Golay codes and the second-order Reed–Muller codes, etc. Cyclic codes have more efficient encoding and decoding algorithms. In this paper, we investigate the optimum distance profiles with respect to the cyclic subcode chains (ODPCs) of the punctured generalized second-order Reed–Muller codes \(\mathcal{GRM }(2,m)^*\) which were applied in Power Control in OFDM Modulations, in channels with synchronization, and so on. For this, two standards are considered in the inverse dictionary order, i.e., for increasing message length. Four lower bounds and upper bounds on ODPC are presented, where the lower bounds almost achieve the corresponding upper bounds in some sense. The discussions are over nonbinary prime fields.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arıkan E.: Channel polarization: a method for constructing capacity-achieving codes for symmetric binaryinput memoryless channels. IEEE Trans. Inf. Theory 55(7), 3051–3073 (2009).

    Google Scholar 

  2. Assmus Jr E.F., Key J.D.: Designs and Their Codes. Cambridge University Press, Cambridge (1992).

  3. Bruen A.: Blocking sets and low-weight codewords in the generalized reed-muller codes. In: Bruen A., Wehlau D., Society C.M. (eds.) Error-Correcting Codes, Finite Geometries, and Cryptography. Contemporary Mathematics, vol. 525. American Mathematical Society, Southport, pp. 161–164 (2010).

  4. Burnashev M., Dumer I.: Error exponents for recursive decoding of Reed–Muller codes on a binary-symmetric channel. IEEE Trans. Inf. Theory 52(11), 4880–4891 (2006).

    Google Scholar 

  5. Cameron P.J., Von Lint J.H.: Graph Theory, Coding Theory and Block Designs. London Mathematical Society. Lecture Note, vol. 19. Cambridge University Press, London (1975).

  6. Chen Y., Han Vinck A.J.: A lower bound on the optimum distance profiles of the second-order Reed–Muller codes. IEEE Trans. Inf. Theory 56(9), 4309–4320 (2010).

    Google Scholar 

  7. Davis J.A., Jedwab J.: Peak-to-mean power control in OFDM, Golay complementary sequences and Reed–Muller codes. IEEE Trans. Inf. Theory 45(7), 2397–2417 (1999).

    Google Scholar 

  8. Delsarte P.: An algebraic approach to the association schemes of coding theory. Philips Res. Rep. Suppl. 10, (1973).

  9. Delsarte P., Goethals J.M.: Alternating bilinear forms over GF(q). J. Comb. Theory. 19(A), 26–50 (1975).

    Google Scholar 

  10. Delsarte P., Goethals J.M., MacWilliams F.J.: On GRM and their relatives. Inf. Contract. 16(5), 403–442 (1970).

    Google Scholar 

  11. Ding C.: The weight distribution of some irreducible cyclic codes. IEEE Trans. Inf. Theory 55(3), 955–960 (2009).

    Google Scholar 

  12. Ding C., Liu Y., Ma C., Zeng L.: The weight distributions of the duals of cyclic codes with two zeros. IEEE Trans. Inf. Theory 57(12), 8000–8006 (2011).

    Google Scholar 

  13. Draper S., Hou X.D.: Explicit evaluation of certain exponential sums of quadratic functions over \({\mathbb{F}}_{p^n}\), p odd. arXiv:0708.3619v1 (2007).

  14. Egawa Y.: Association schemes of quadratic forms. J. Comb. Theory. 38(A), 1–14 (1985).

    Google Scholar 

  15. Elias P.: List decoding for noisy channels. Technical Report 335, Research Laboratory of Electronics, MIT (1957).

  16. Feng K., Luo J.: Weight distribution of some reducible cyclic codes. Finite Fields Appl. 14(2), 390–409 (2008).

    Google Scholar 

  17. Geil O.: On the second weight of generalized Reed–Muller codes. Des. Codes Cryptogr. 48, 323–330 (2008).

    Google Scholar 

  18. Han Vinck A.J., Luo Y.: Optimum distance profiles of linear block codes. In: Proceedings of IEEE International Symposium on Information Theory, pp. 1958–1962 (2008).

  19. Helleseth T., Kløve T., Mykkeltveit J.: The weight distribution of irreducible cyclic codes with block lengths \(n_1((q^l-1)/N)\). Discret. Math. 18, 179–211 (1977).

  20. Holma H., Toskala A.: WCDMA for UMTSHSPA Evolution and LTE, 4th edn. Wiley, London (2007).

  21. Kacewicz A., Wicker S.: Application of Reed–Muller codes for localization of malicious nodes. In: Proceedings of IEEE International Conference on Communications (ICC’10), Capetown (2010).

  22. Kasami T., Lin S., Peterson W.: New generalization of Reed–Muller codes. Part I: primitive codes. IEEE Trans. Inf. Theory 14, 189–199 (1968).

    Google Scholar 

  23. Korada S.B., Şaşoğlu E., Urbanke R.: Polar codes: characterization of exponent, bounds, and constructions. IEEE Trans. Inf. Theory 56(12), 6253–6264 (2010).

    Google Scholar 

  24. Lidl R., Niederreiter H.: Finite Fields. Cambridge University Press, Cambridge (1997).

  25. Liu X., Luo Y.: The weight distributions of some cyclic codes with three or four nonzeros over \({\mathbb{F}}_3\). arXiv:1302.0394v1.

  26. Liu X., Luo Y., Shum K.W.: On the optimum cyclic subcode chains of \({\cal RM}(2,m)^*\) for increasing message length. arXiv:1306.0710v1.

  27. Luo Y., Han Vinck A.J.: On a classification of cyclic subcode chains. In: 2009 International Conference on Communications and Networking in China (ChinaCom-2009), Xi’an (2009).

  28. Luo Y., Han Vinck A.J., Chen Y.: On the optimum distance profiles about linear block codes. IEEE Trans. Inf. Theory 56(3), 1007–1014 (2010).

    Google Scholar 

  29. Luo J., Tang Y., Wang H.: Cyclic codes and sequences: The generalized Kasami case. IEEE Trans. Inf. Theory 56(5), 2130–2142 (2010).

    Google Scholar 

  30. MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1988).

  31. McEliece R.J.: Irreducible cyclic codes and Gauss Sums. In: Combinatorics, Part I: Theory of Designs, Finite Geometry and Coding Theory, vol. 55, pp. 179–196. Mathematical Centre Tracts, Mathematisch Centrum, Amsterdam (1974).

  32. Paterson K.G.: Generalized Reed–Muller codes and power control in OFDM modulation. IEEE Trans. Inf. Theory 46(1), 104–120 (2000).

    Google Scholar 

  33. Sloane N.J.A.: An introduction to association schemes and coding theory. In: Askey R. (ed.) Theory and Application of Special Functions, pp. 225–260. Academic Press, New York (1975).

  34. Tanner R., Woodard J.: WCDMA—Requirements and Practical Design. Wiley, London (2004).

  35. van Dijk M., Baggen S., Tolhuizen L.: Coding for informed decoders. In: Proceedings of IEEE International Symposium on Information Theory, p. 202. Washington, DC (2001).

  36. Weldon Jr E.J.: New generalizations of the Reed–Muller codes. Part II: nonprimitive codes. IEEE Trans. Inf. Theory 14(3), 199–205 (1968).

    Google Scholar 

  37. Wozencraft J.: List Decoding. Technical Report 48, pp. 90–95. Quarterly Progress Report, Research Laboratory of Electronics, MIT (1958).

Download references

Acknowledgments

This work was supported in part by the National Basic Research Program of China under Grants 2012CB316100, 2013CB338004, and the National Natural Science Foundation of China under Grant 61271222. We would also like to acknowledge the foundation TS0520103001 of Shanghai Jiao Tong University and University of Leuven.

Author information

Authors and Affiliations

  1. Department of Computer Science and Engineering, Shanghai Jiao Tong University, 800 Dongchuan Road, Min Hang District, Shanghai, 200240, China

    Xiaogang Liu & Yuan Luo

  2. National Mobile Communications Research Laboratory, Southeast University, Nanjing, China

    Yuan Luo

Authors
  1. Xiaogang Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yuan Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yuan Luo.

Additional information

Communicated by D. Jungnickel.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Liu, X., Luo, Y. On the bounds and achievability about the ODPC of \(\mathcal{GRM }(2,m)^*\) over prime fields for increasing message length. Des. Codes Cryptogr. 74, 533–557 (2015). https://doi.org/10.1007/s10623-013-9877-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-013-9877-5

Keywords

Mathematics Subject Classification

Search

Navigation